Lieb's theorem – In 1989, Lieb [1]} proved that for the attractive Hubbard model,
with \(U<0\)  (and allowing more generally for site-dependent \(U^{\vphantom{\dagger }}_i<0\ \forall \ i\) ) that if the total
number of electrons \(N\)  is even, then the ground state of \({\hat{H}}\)  is unique, and is a total spin singlet,
i.e. with \(S=0\) . A corollary is that if \(U>0\)  (independent of \(i\) ), \(\Lambda \)  is bipartite with \(|\textsf {A}|\ge |\textsf {B}|\) , and \(N = |\Lambda |\)  is even, then the ground state of \({\hat{H}}\)  has total spin \(S={\textstyle {1 \over 2}}\big (|\textsf {A}|-|\textsf {B}|\big )\)  and degeneracy \(2S+1\)  (i.e. spin degeneracy is the only degeneracy). A convenient example is the so-called Lieb lattice, in which the \(\textsf {A}\)  sublattice is a square lattice and there is a \(\textsf {B}\)  sublattice site at the center of every link.
Lieb's proof extends the Perron-Frobenius argument deployed in one-dimensional magnetic systems [2]} by invoking a tool known as “spin-space reflection positivity". The theorem then entails a hierarchy
of lowest energy levels in different total spin sectors, \(E^{\vphantom{\dagger }}_0(S) < E^{\vphantom{\dagger }}_0(S+1)\) , where \(S\in \big \lbrace {\textstyle {1 \over 2}}(|\textsf {A}|-|\textsf {B}|),\ldots ,{\textstyle {1 \over 2}}|\Lambda |-1\big \rbrace \) . For bipartite lattices, the conditions of Lieb's theorem guarantee a ferromagnetic ground state when there is a sublattice imbalance, i.e. \(|\textsf {A}|\ne |\textsf {B}|\) .
